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# Exponential Functions: math.exp()

**00:00**
In this lesson, we’ll talk about general exponential functions and then focus on the exponential function defined in the `math`

module.

**00:09**
Before we jump into what an exponential function is, let’s review real quick what a power function is. In power functions, the base is what changes and the power or exponent is fixed.

**00:22**
When the exponent is a positive integer, these functions are usually called polynomial functions. On the other hand, exponential functions are power functions, but here, the base is fixed and it’s the exponent that changes. A lot of times these are written as, say, *a* to the power of *x*. *a* is the base—, that’s fixed—and it’s the power *x* that’s changing.

**00:48**
The fixed base *a* is a positive number in most applications. Otherwise, you’re going to have to deal with complex numbers. Qualitatively, when *a*, the base, is greater than 1, the values of the function increase as *x* increases. So, in other words, if this base *a* is greater than 1, as we make this exponent larger, the entire output of the function also increases. On the other hand, if *a* is between 0 and 1, then the values of the output of the function—they’re going to decrease as the input *x* increases.

**01:25**
There’s an important exponential function, which a lot of times it’s called *the* exponential function, and this is one where the base *a* is Euler’s number *e*, which we’ve encountered before.

**01:37**
So, *e* is around 2.71828, and so if that constant *a* is exactly Euler’s number, a lot of times, this is called the exponential function or the natural exponential function.

**01:52**
Because of its importance, the `math`

module provides an implementation of the exponential function, and the name of the function is `exp()`

.

**02:00**
It takes in one input argument *x*, which is any float or any integer.

**02:06**
The exponential function is important in many applications involving exponential growth or decay. We’ll take a look at an example in the next lesson of this, but for now, let’s just get comfortable with the exponential function.

**02:20**
Let’s just compute a couple values of the exponential function. So, say, evaluated at `3`

or evaluated at, say, `-3`

. And so, here, the input can be any float, any number. It doesn’t matter whether it’s negative or positive.

**02:36**
The exponential is defined for every real number. Now, the exponential function being *e* to the power of *x* means that if we evaluate the exponential function at `1`

, it should be pretty close to the value of `e`

in the `math`

module.

**02:55**
*e* to the power of 1—so in other words, the exponential function evaluated at `1`

—*e* to the 1 is *e*, so we should either get `True`

or we can test this with the `isclose()`

function.

**03:07**
But go ahead and see what happens when you do this. We get `True`

. Python’s doing a good job in being consistent that *e* to the power of 1 is exactly *e*, and so the value in the `e`

constant in the `math`

module does evaluate exactly to the exponential function evaluated at `1`

.

**03:27**
Now, the exponential function is just a power function, so if I compute the power of `e`

to the power of `2`

, that should be pretty close to the exponential function evaluated at `2`

. Let’s see what Python does in this case.

**03:47**
We get `False`

. But you know that these are supposed to be pretty close—let’s see how close they are. Let’s just see if the `isclose()`

function will return a `True`

value.

**03:58**
Go ahead and try: Is the value of the constant `e`

as stored in the `math`

module raised to the power of `2`

—is that close to the value of the exponential function evaluated at `2`

? And we get `True`

.

**04:17**
And if you wanted to take a look at these values a little bit more so that we can see them, let’s just print these out to the console.

**04:27**
They’re pretty close and definitely within `10`

to the power of `-9`

in terms of the relative size of those values.

**04:35**
In the next lesson, we’ll take a look at how the exponential function is used to model the decay of a radioactive substance.

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